Transverse modes

Transverse modes are the spatial field patterns that an optical cavity or waveguide supports perpendicular to the propagation direction. Each transverse mode has a distinct spatial intensity distribution, and the full optical field is generally a superposition of multiple transverse modes — though good single-mode laser cavities are designed to support only the fundamental transverse mode.

Hermite-Gaussian modes (TEM_mn). For laser cavities with rectangular symmetry (and most stable resonators), the transverse mode set is the Hermite-Gaussian series:

$$\text{TEM}_{mn} \;\propto\; H_m\!\left(\frac{\sqrt{2}x}{w}\right) H_n\!\left(\frac{\sqrt{2}y}{w}\right) e^{-(x^2 + y^2)/w^2},$$

where $H_m$ is the Hermite polynomial of order $m$ and $w$ is the beam waist. The integer indices $(m, n)$ count the number of nodes in the field pattern along the x and y axes respectively.

Standard mode profiles.

Mode	Description
TEM₀₀	Gaussian; bell-shaped intensity; no nodes
TEM₁₀	Two lobes side-by-side, one node vertically
TEM₀₁	Two lobes stacked, one node horizontally
TEM₁₁	Four-lobe pattern, one node in each direction
TEM₂₀	Three lobes vertically (or horizontally)
TEM₃₀	Four-lobe row

Laguerre-Gaussian modes (LG_pl). For cylindrically-symmetric cavities, the natural mode set is Laguerre-Gaussian:

$$\text{LG}_{pl} \;\propto\; \left(\frac{r}{w}\right)^{|l|} L_p^{|l|}\!\left(\frac{2r^2}{w^2}\right) e^{-r^2/w^2} e^{i l \phi},$$

with radial index $p$ (= 0, 1, 2, ...) and azimuthal index $l$ (= ..., -2, -1, 0, 1, 2, ...). The $l$-index carries orbital angular momentum: $l = 0$ has no angular momentum, $l = 1$ carries one $\hbar$ per photon, etc.

Operating frequency depends on transverse mode order. Different transverse modes of the same cavity have slightly different resonant frequencies. For a hemispherical or two-mirror cavity, the resonance condition is:

$$\nu_{q,m,n} \;=\; \frac{c}{2L}\left[q + (m+n+1) \frac{\arccos(g_1 g_2)^{1/2}}{\pi}\right],$$

where $q$ is the longitudinal mode index and $g_1, g_2$ are the cavity stability parameters. Higher-order transverse modes are typically separated from TEM₀₀ by ~100 MHz to 100 GHz depending on cavity geometry.

Why higher-order modes are undesirable in most lasers. TEM₀₀ has several practical advantages:

• Beam quality: TEM₀₀ has $M^2 = 1$ (diffraction-limited); higher-order modes have $M^2 > 1$ and diverge faster
• Focusability: TEM₀₀ focuses to the smallest spot achievable for a given wavelength
• Single-mode operation in fiber: TEM₀₀ couples efficiently to single-mode fiber's fundamental mode; higher-order modes do not
• Predictable beam profile: TEM₀₀ has a known Gaussian profile; higher-order modes vary more from cavity to cavity

Most commercial lasers are designed for TEM₀₀ operation by appropriately shaping the gain region or by including an intracavity aperture that selectively blocks higher-order modes.

Multimode laser operation. Some applications deliberately operate at higher-order modes:

• High-power industrial lasers: TEM₀₀ may not give enough power; multimode operation provides higher total power at lower beam quality
• Multimode pump lasers for fiber lasers: large multimode pump diodes have $M^2 \gg 1$ but couple acceptably to large-mode-area pump fibers
• Donut beams for STED microscopy: LG modes with $l \neq 0$ provide the donut excitation pattern
• Optical tweezers with structured light: higher-order modes provide complex potential landscapes for trapped particles

Transverse modes in waveguides. Optical waveguides (fibers, integrated waveguides) support a discrete set of transverse modes determined by the waveguide geometry. The mode labeling differs from laser cavities:

• Fiber modes: LP modes (linearly-polarized approximation) labeled LP01, LP11, LP21, LP02, etc. The LP01 mode is the fundamental and analog of TEM₀₀.
• Slab waveguide modes: TE_m and TM_m where $m$ counts the number of nodes across the slab
• Channel waveguide modes: TE₀, TM₀, TE₁, etc. with full 2D mode structure

Beam quality of multi-mode superpositions. When a beam contains multiple transverse modes, $M^2 > 1$. For Hermite-Gaussian modes:

$$M^2 \;=\; (2m + 1)(2n + 1) \quad \text{(separable case)},$$

so TEM₁₀ has $M^2 = 3$, TEM₁₁ has $M^2 = 9$. A multimode laser containing modes up to TEM₃₃ would have $M^2 \approx 49$.

Transverse mode beating. Two transverse modes simultaneously present at a photodetector beat at their frequency difference, producing RF noise. For external-cavity lasers, transverse mode beat notes are at characteristic frequencies determined by cavity geometry, providing a diagnostic for transverse mode purity.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 3 (beam optics) and Ch. 11 (laser resonators); Siegman, Lasers (University Science Books, 1986), Ch. 16 — the comprehensive treatment of laser-cavity transverse modes; Andrews & Phillips, Laser Beam Propagation through Random Media for atmospheric effects on transverse mode structure.